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Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroids

Panagiotis Batakidis, Fani Petalidou

Abstract

We construct an algebra and a complex of multidifferential operators on tensor products of a Courant algebroid E with values in the endomorphism bundle of a smooth vector bundle B, predual of E, extending the standard complex of the Courant-Dorfman algebra of E. Also, we study Dorfman connections of E on B, and show that the Cartan calculus, curvatures of induced connections and basic differential geometric identities of them make sense in this algebra.

Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroids

Abstract

We construct an algebra and a complex of multidifferential operators on tensor products of a Courant algebroid E with values in the endomorphism bundle of a smooth vector bundle B, predual of E, extending the standard complex of the Courant-Dorfman algebra of E. Also, we study Dorfman connections of E on B, and show that the Cartan calculus, curvatures of induced connections and basic differential geometric identities of them make sense in this algebra.

Paper Structure

This paper contains 16 sections, 21 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Courant algebroids, Dirac structures, and their cohomologies
  3. Courant algebroids and Dirac structures
  4. Cohomologies of a Courant algebroid
  5. Universal enveloping and convolution algebra
  6. The cohomology of the Courant-Dorfman algebra of a Courant algebroid
  7. The cohomology of the Courant-Dorfman algebra of differential operators of a Courant algebroid
  8. Dorfman connections
  9. Predual vector bundle
  10. Dorfman connections
  11. Curvature of Dorfman connections
  12. Induced connections and Bianchi identity
  13. Connection on the dual bundle
  14. Connection on the endomorphism bundle
  15. Connections on modules of differential operators
  16. ...and 1 more sections

Key Result

Theorem 1.1

For a Courant algebroid $E$, (i) the differential, contractions and Cartan calculus in $\mathcal{C}(\mathcal{E};\mathcal{R})$, extend to analogous operators and calculus in $\mathfrak{D}(\mathcal{E};\mathcal{R})$ and make it a cochain complex; (ii) the projection map from $(\mathfrak{D}(\mathcal{E};

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.5: Courant algebroid over a point
  • Example 2.6: Standard Courant algebroid
  • Example 2.7: The double of a Lie bialgebroid
  • Example 2.8
  • Definition 2.9
  • ...and 40 more